Methods of Hankel and Toeplitz Operators in Noncommutative Analysis

主要研究者：弗拉基米尔佩勒提案号0200712摘要主要研究者将在非交换分析中使用向量汉克尔和Toeplitz算子。 尽管矩阵和算子函数的逼近理论最近取得了进展，但在控制理论、多变量平稳过程等应用中仍有许多重要的开放问题。主要研究者将寻求有理矩阵函数的超最优逼近阶的精确估计，以及Toeplitz和Wiener-Hopf算子作用于向量函数空间的预解式的精确估计，研究了矩阵函数的Wiener-Hopf分解、主题分解和标准分解的性质，研究了多元平稳过程在谱项下的不同正则性条件，研究了Hankel-Schur乘子，研究了由Schrodinger算子产生的某些积分算子的Schatten-von Neumann性质。这些是Hankel算子和Toeplitz算子。近20年来，这类算子的理论得到了迅速的发展，近年来，为了满足控制理论、预测理论和其它应用的需要，研究向量函数空间上的Toeplitz和Hankel算子已成为必要。这导致了一个相当复杂的理论，主要研究人员最近获得了有趣的结果，需要一种新的技术。他将继续朝这个方向努力，并将Hankel和Toeplitz算子应用于向量函数空间中的不同问题的非对易分析。

PI: Vladimir PellerProposal Number 0200712AbstractThe principal investigator will use vectorial Hankel and Toeplitz operators in noncommutative analysis. Inspite of the recent progress of approximation theory for matrixand operator functions there are still many open problemsimportant in applications such as control theory, multivariatestationary processes, etc. The principal investigator will look for a sharp estimate of the degree of the superoptimalapproximant of a rational matrix function and sharp estimates ofthe resolvents of Toeplitz and Wiener-Hopf operators acting onspaces of vector functions, study properties of Wiener-Hopf,thematic, and canonical factorizations of matrix functions, studydifferent regularity conditions for multivariate stationaryprocesses in spectral terms, study Hankel-Schur multipliers, studySchatten--von Neumann properties of certain integral operatorsarising from Schrodinger operators.Among operators acting on spaces of analytic functions there aretwo classes that play an extremely important role. These areHankel operators and Toeplitz operators. The theory of suchoperators has been rapidly developing for the last 20 years.Recently it has become clear that to satisfy the needs of controltheory, prediction theory and other applications, it is necessaryto study Toeplitz and Hankel operators on spaces of vectorfunctions. This leads to a considerably more complicated theory,The principal investigator has obtained recently interestingresults that required a new technique. He is going to continue towork in this direction and apply Hankel and Toeplitz operators onspaces of vector functions to different problems in noncommutativeanalysis.